Why can I not subtract objects from each other?
Your problem is likely coinciding faces.
Boolean operations do not work well on mesh objects with overlapping, co-planar or almost parallel faces.

My advice is to make it so that the "wireframe object" is slightly bigger than the "solid one", making it protrude a little to the outside, while maintaining whatever precise measurement is indented, so that the outer faces are not exactly overlapping the larger ones.

Optionally go to the Properties Window > Object > Display > Maximum Draw Type and change it from Textured to Wire for the "edges object" you are trying to subtract, so that you can more clearly see the finished operation in the viewport.